US 7304736 B2
A method of nonlinear polarimetry for measuring higher order moments of the E field of an optical signal is provided. The method includes imposing a phase delay on a first polarization of a received optical signal with respect to a second polarization of the optical signal to produce an intermediate optical signal having a time varying polarization. A polarization of the intermediate optical signal is suppressed. The intermediate optical signal is detected with a plurality of photodetectors, with at least one photodetector configured to be responsive to a nonlinear optical process. Spectra of the photodetector outputs are calculated to determine higher order moments of the E field, and the moments are transformed to obtain the polarization measurement.
1. A method of nonlinear polarimetry for measuring higher order moments of the E field of an optical signal, comprising the steps of:
imposing a phase delay on a first polarization of a received optical signal with respect to a second polarization of the optical signal to produce an intermediate optical signal having a time varying polarization;
suppressing a polarization of the intermediate optical signal;
detecting the intermediate optical signal with a plurality of photodetectors, at least one photodetector being configured to be responsive to a nonlinear optical process;
calculating spectra of the photodetector outputs to determine higher order moments of the E field; and
transforming the higher order moments to obtain the polarization measurement.
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8. A method of nonlinear polarimetry for measuring higher order moments of an E field of an optical signal, comprising the steps of:
coupling an optical signal to an optical waveguide having a plurality of waveguide birefringences;
imposing a phase delay on a first polarization of the optical signal with respect to a second polarization of the optical signal;
scattering the light with polarization sensitive gratings;
detecting the scattered light from a plurality of gratings with a plurality of photodetectors, at least one photodetector being configured to be responsive to a nonlinear optical process, the detecting generating detector signals; and
transforming the detector signals to polarization measurements.
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This Application is a Divisional of prior application Ser. No. 10/413,962 filed on Apr. 15, 2003, now U.S. Pat. No. 7,079,246 currently pending, to Paul S. Westbrook. The above-listed Application is commonly assigned with the present invention and is incorporated herein by reference as if reproduced herein in its entirety under Rule 1.53(b).
This invention relates to a method and apparatus for measuring the polarization of light.
High-speed optical fiber communication systems operate by encoding information (data) onto lightwaves that typically propagate along optical fiber paths. Most systems, especially those used for medium to long distance transmission employ single mode fiber. As implied by the name, single mode fibers propagate only one mode of light below cutoff. The single mode typically includes many communications channels. The communications channels are combined into the one transmitted mode, as by wavelength division multiplexing (WDM) or dense wavelength division multiplexing (DWDM).
While only one mode is transmitted, that mode actually comprises two perpendicular (orthogonal) polarizations. The polarization of these two components varies undesirably as the waves propagate along a fiber transmission path. The distortion of the optical signals caused by the varying polarization is called polarization mode dispersion (PMD). PMD can be corrected through a combination of measurements of the PMD and the control of active corrective optics.
Polarimeters measure the polarization of light. Polarimeters can generate signals representing a measured degree of polarization that can be useful for diagnostic purposes. The signals can also be advantageously used for polarization correction using feedback techniques to minimize PMD.
Polarimeters generally employ one or more photodetectors and related electro-optical components to derive basic polarization data. The raw photodetector signal measurements are typically transformed by mathematical techniques into standard polarization parameters. In the prior art, the photodetector outputs are generally averaged, as by some electronic time constant, and then multiplied as part of the signal processing and transformation process. The problem with averaging at detection is that instantaneous temporal information lost through averaging cannot be retrieved later.
What is needed for more accurate polarization measurements is a polarimeter that instantaneously measures polarimeter photodetector outputs without averaging, multiplies the unaveraged signals early in signal processing, and then averages and transforms the signals into polarimetry parameters.
An improved method and apparatus for the measurement of the polarization of light uses nonlinear polarimetry. The higher order moments of the E field are measured and then transformed into standard polarimetry parameters yielding the polarization of the light. In a first embodiment, the light to be measured is transmitted through a rotating retarder capable of rotating at a plurality of angles with at least two retardances Δ. The retarder is optically coupled to a fixed analyzer. The light from the analyzer is then detected by linear and nonlinear photodetectors. The spectra from the detectors is calculated and transformed, to obtain the polarization. In a second embodiment, the light to be measured is received by an optical fiber comprising a plurality of fiber birefringences to retard the light. Polarization sensitive gratings along the length of the fiber scatter the light, and photodetectors detect the scattered light. The signals from the photodetectors can then be transformed to obtain the polarization.
Apparatus in two preferred embodiments can perform the inventive method. In the first embodiment, nonlinear and linear photodetectors are preceded by a rotating retarder, rotating at a plurality of angles with a retardance, and an analyzer, such as a fixed polarizer. In the second preferred apparatus, a plurality of photodetectors are located adjacent to polarization sensitive gratings situated in a birefringent optical waveguide located between each of the polarization sensitive gratings.
The advantages, nature and various additional features of the invention will appear more fully upon consideration of the illustrative embodiments now to be described in detail in connection with the accompanying drawings. In the drawings:
It is to be understood that the drawings are for the purpose of illustrating the concepts of the invention, and except for the graphs, are not to scale.
This description is divided into two parts. Part I describes the inventive method for polarimetry and two embodiments for making polarization measurements according to the inventive method. For those skilled in the art, Part II further develops, defines, and introduces the concepts of invariance, state of polarization and degree of polarization, and the foundation equations governing nonlinear polarimetry as best understood by applicants at the time of the invention.
Part I: Nonlinear Polarimetry
Standard polarimeters use linear detectors and thus measure terms quadratic in the E field. These can be considered 2nd order moments of the E field and are related to the power and the Stokes parameter of the E-field. A detector measuring intensity squared, though would measure 4th order moments of the E field. Such higher order moments can have more information about the E-field. Simply put, a higher order moment of some time varying quantity is simply the time average of a higher power of the quantity. The first power is always just the mean. The second power is the standard deviation and so on.
Nine moments of the E field can be measured with apparatus 10 as shown in
Retarder 12 is an optical component that retards one polarization with respect to the orthogonal polarization. In terms of E fields, the retarder gives one of the polarizations a phase with respect to the other orthogonal E component. Examples are ½ λ or ¼ λ retarders.
A ¼ λ wave retarder causes a
Similarly, a ½ λ wave retarder causes a delay difference of π:
Here, the retarder 12 is a generic retarder. It has an arbitrary angle and arbitrary phase retardance. The angle C sets the two linear states of polarization on which the phase difference Δ is applied.
Analyzer 13 is a polarizer. It passes the light of polarization A, and suppresses all other polarizations. By rotating analyzer 13, light of polarization A is a continuous sampling of all 2π polarizations. A detector viewing the light output of a continuously rotating analyzer registers a periodic waveform. The Fourier spectra of that waveform contains a DC component (near 0), and all other components of the spectra.
A preferred alternative version of this embodiment rotates retarder 12, with a fixed analyzer 13 to generate the sine and cosine quadrature components of the Fourier spectra of the detector output. These components yield the nine E field higher order components.
The response of the nonlinear detector is:
The nonlinear detector would measure I2, and the filter in the DC electronics would determine an averaging time, as in the linear case:
All nine components can be measured if one rotates both analyzer A and retarder C in a manner analogous to the linear Stokes case. By performing measurements at the different sum and difference frequencies proportional to nine linearly independent superpositions of
For the nonlinear polarimeter one would toggle between:
A static measurement of the moments can also be done with apparatus 20 as shown in
Here each grating with its nonlinear detector 24 will generate an output signal which is proportional to a linear transformation of the Stokes parameters. Each detector 24 signal is linearly related to a Stokes tensor component. Therefore with proper grating 22 alignments, the nine detector 24 outputs have a linear relationship with the nine Stokes tensor components. Gratings 22 are each aligned in different directions. Gratings 22 are each aligned azimuthally about the axis of the optical fiber. Both the grating 22 alignments and birefringences are aligned such that the 9×9 calibration matrix is invertible.
The measured moments have several uses. The degree of polarization (DOP) is most useful with the Stokes vector because it does not depend on the SOP. That is you can bump the fiber, and the DOP will not change. In other words, the DOP is invariant (see definition of invariant later in Part II) under unitary or lossless transformations. This makes it valuable as a monitoring quantity since a fiber bump does not change it, at least not as much as a bump causes a change in S1 or S2. The higher order moments also have invariants. To understand the invariance of
But, there are more terms, since
Using such an embodiment, one can measure aVl 2−Vl, where a is such that when the signal is constant, aV1 2−Vn=0, then Vn−aVl 2≧0, since intensity fluctuations always make
An important advantage to having both a linear and nonlinear detector is that the nonlinear detector can be “nonlinearized” by subtracting out the linear part. This is illustrated by
Actual fabrication forms and techniques suitable for constructing the inventive apparatus in general, includes, but is not limited to, bulk optical components, optical fibers and optical fiber components, and integrated techniques, including planer waveguides, and other integrated optical components.
Part II: Theoretical Development of Nonlinear Polarimetry Including the Definition of Invariance
Invariance: A polarization transformation is said to be invariant when there is a polarization transformation in which the two principle states are delayed by less than the coherence length of the light. This is an invariant transformation. In mathematical terms:
∫dtE1(t)E2(t+τc)≠0, τc=correlation time, E1, E2 are principal states, and τinvariant<<τc. In short: Invariant=unitary with τ<τc where τ is the maximum time delay between polarization components. Also the ratio of the two principle states must remain fixed, i.e., the “fiber touch” cannot be before a large PMD element such as a fiber link, since changing the launch polarization into a fiber with PMD will change the ratio of the two principle states and hence alter the output pulse shape and its higher order moments. The “fiber touch” that we wish to avoid being sensitive to through the use of invariants is that directly before the polarization monitor. With standard polarimeters the only invariants are the total power and the DOP.
State of polarization and Degree of Polarization: It is useful to provide a clear definition of “state of polarization” (or SOP), with respect to an optical signal propagating through a fiber. In general, if the core-cladding index difference in a given optical fiber is sufficiently small, then the transverse dependence of the electric field associated with a particular mode in the fiber may be written as:
In accordance with the teachings of the present invention, the state of polarization (SOP) of an optical fiber will be described using the Jones calculus and the Stokes parameters, since these are both complete and commonly used. The Jones vector J that describes the field at any location z or point in time t is given by the following:
A more complete description of the state of polarization is based on the defined Stokes parameters, since this method also accounts for the degree of polarization (DOP) of a non-monochromatic signal. In terms of the Jones vector parameters, the four Stokes parameters are defined by:
It has been recognized in accordance with the teachings of the present invention that the full state of polarization (SOP) cannot be determined by merely evaluating the signal passing through a single polarizer. Birefringence alone has also been found to be insufficient. In particular, a polarimeter may be based on a presumption that the optical signal to be analyzed is passed through a compensator (birefringent) plate of relative phase difference Γ with its “fast” axis oriented at an angle C relative to the x axis (with the light propagating along the z direction). Further, it is presumed that the light is subsequently passed through an analyzer with its transmitting axis oriented at an angle A relative to the x axis. Then, it can be shown that the intensity I of the light reaching a detector disposed behind the compensator and analyzer can be represented by:
Following from the equations as outlined above, a polarimeter may be formed using a compensator (for example, a quarter-wave plate), a polarizer, and a detector. In particular, the following four measurements, used in conventional polarimeters, unambiguously characterize the Stokes parameters:
Standard polarimeters measure the degree of polarization (DOP), or Stokes parameters that represent the polarization, by taking time averaged measurements of the x and y components of the E-field as represented by:
But, higher order moments can be measured as well as:
A nonlinear polarimeter is a device that measures the higher order moments. These measurements can provide extra information about the bit stream or any polarized or partially polarized signal.
The number of moments that can be measured can be determined in two ways. The E-field representation as mentioned above is one way:
Define (m, n) where m=#Ex's and n=#Ey's
This gives 1×(4,0)+1×(0,4)+2(1,3)+2(3,1)+3(2,2)=9 or
Alternatively, the un-averaged Stokes products SiSj can be constructed. These are the 2nd order moments before averaging:
Also, S0S0 is not independent, because before averaging DOP=1, therefore, before time averaging, S0S0=S1S1+S2S2+S3S3.